Ideal of a Lie ring: Difference between revisions

Latest revision as of 16:12, 17 July 2009

This article describes a Lie subring property: a property that can be evaluated for a subring of a Lie ring
View a complete list of such properties
VIEW RELATED: Lie subring property implications | Lie subring property non-implications | Lie subring metaproperty satisfactions | Lie subring metaproperty dissatisfactions | Lie subring property satisfactions |Lie subring property dissatisfactions

ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie subring property analogous to the subgroup property: normal subgroup
View other analogues of normal subgroup | View other analogues in Lie rings of subgroup properties (OR, View as a tabulated list)

Definition

A subset $B$ of a Lie ring $A$ is termed an ideal in $A$ if $B$ is additively a subgroup and $[x, y] \in B$ whenever $x \in B$ and $y \in A$ .

Relation with properties in related algebraic structures

Lie algebra

Further information: Ideal of a Lie algebra

A Lie algebra is a Lie ring that is simultaneously (i.e., with the same operations) an algebra over a field. An ideal of a Lie algebra is an ideal of the underlying Lie ring that is also a linear subspace, i.e., it is closed under multiplication by scalars in the field.

Ring whose commutator operation is the Lie bracket

Suppose $R$ is an associative ring: an abelian group with a distributive associative multiplication. We can define the Lie ring associated with $R$ as $R$ with the same addition and wit hthe Lie bracket given by the commutator operation $[x, y] = x y - y x$ .

Then, the property of being an ideal of the Lie ring is equivalent to the property of being a Lie ideal in $R$ . Being a Lie ideal is weaker than being a two-sided ideal. It is incomparable with the property of being a left ideal or being a right ideal. Moreover, a subset that is both a left ideal and a Lie ideal is two-sided ideal. Similarly, a subset that is both a right ideal and a Lie ideal is a two-sided ideal.

Group via the Lazard correspondence

Suppose $G$ is a Lazard Lie group and $L$ is its Lazard Lie ring. Under the natural bijection from $L$ to $G$ , the ideals of $L$ correspond to the normal subgroups of $G$ .

Relation with other properties

Stronger properties

Derivation-invariant Lie subring

Weaker properties

@@ Line 1: / Line 1: @@
 {{Lie subring property}}
-{{analogue in-of|Lie ring|subgroup|normality}}
+{{analogue of property|
+old generic context = group|
+old specific context = subgroup|
+new generic context = Lie ring|
+new specific context = Lie subring|
+old property = normal subgroup}}
 ==Definition==
 A subset <math>B</math> of a [[Lie ring]] <math>A</math> is termed an ''ideal'' in <math>A</math> if <math>B</math> is additively a subgroup and <math>[x,y] \in B</math> whenever <math>x \in B</math> and <math>y \in A</math>.
+==Relation with properties in related algebraic structures==
+===Lie algebra===
+{{further|[[Ideal of a Lie algebra]]}}
+A [[Lie algebra]] is a Lie ring that is simultaneously (i.e., with the same operations) an algebra over a field. An ideal of a Lie algebra is an ideal of the underlying Lie ring that is also a linear subspace, i.e., it is closed under multiplication by scalars in the field.
+===Ring whose commutator operation is the Lie bracket===
+Suppose <math>R</math> is an associative ring: an abelian group with a distributive associative multiplication. We can define the Lie ring associated with <math>R</math> as <math>R</math> with the same addition and wit hthe Lie bracket given by the commutator operation <math>[x,y] = xy - yx</math>.
+Then, the property of being an ideal of the Lie ring is equivalent to the property of being a Lie ideal in <math>R</math>. Being a ''Lie ideal'' is weaker than being a two-sided ideal. It is incomparable with the property of being a left ideal or being a right ideal. Moreover, a subset that is both a left ideal and a Lie ideal is two-sided ideal. Similarly, a subset that is both a right ideal and a Lie ideal is a two-sided ideal.
+===Group via the Lazard correspondence===
+Suppose <math>G</math> is a [[Lazard Lie group]] and <math>L</math> is its [[Lazard Lie ring]]. Under the natural bijection from <math>L</math> to <math>G</math>, the ideals of <math>L</math> correspond to the [[normal subgroup]]s of <math>G</math>.
+==Relation with other properties==
+===Stronger properties===
+* [[Weaker than::Derivation-invariant Lie subring]]
+===Weaker properties===
+* [[Stronger than::Lie subring whose sum with any subring is a subring]]
+* [[Stronger than::Sub-ideal of a Lie ring]]